3
$\begingroup$

The set $A=\{f:[0,1] \rightarrow \mathbb{R} : f$ is continuous$\}$ is a ring with the standard function addition and multiplication. Which are the prime ideals in $A$?

The only thing I've managed to observe is that the units of this ring are the functions that never take the value $0$, so every element of a prime ideal $P$ takes the value $0$ at least once.

$\endgroup$
  • $\begingroup$ did you figure out which are the ideals in that ring? $\endgroup$ – Jorge Fernández Hidalgo Feb 13 '15 at 5:23
  • $\begingroup$ I suspects the ideals are of the form $\{f \in A: f(x)=0 \forall x \in S\}$, where $S$ is a subset of $[0,1]$ $\endgroup$ – Marco Flores Feb 13 '15 at 6:24
  • $\begingroup$ The maximal ideals are well-known. But what about the prime ideals? good question. $\endgroup$ – Mister Benjamin Dover Feb 14 '15 at 17:02
1
$\begingroup$

fix $x_0$ ,then ideal $I_0=\{f\in A: f(x_0)=0\}$ is prime. For that if $fg\in I_0$ Then either $f$ or $g$ must be zero at $x_0$.

Edit1:i dont know if these are all ...

Edit 2:https://mathoverflow.net/questions/35793/prime-ideals-in-c0-1

$\endgroup$
  • $\begingroup$ I can see those are prime ideals.. but, why are they all the prime ideals? $\endgroup$ – Marco Flores Feb 13 '15 at 6:01
  • $\begingroup$ You get that every element $f$ of a proper ideal $I$ satisfies $f(x)=0$ for some $x$, but that $x$ need not be the same for every $f \in I$, or does it? $\endgroup$ – Marco Flores Feb 13 '15 at 6:09
  • $\begingroup$ Those are all maximal ideals. $\endgroup$ – user26857 Feb 13 '15 at 7:06
  • $\begingroup$ And so prime ideals.but apparently there are examples of prime ideals of C(I) that arent maximal. $\endgroup$ – BigM Feb 13 '15 at 7:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.